The absolute value is alone on one side of the equation, which means it is ready to be set equal to the positive and negative value of the other side of the equation while removing the absolute value signs.

You should get

and

Distribute the 2 in each equation to both the x and the 4 to get

and

Subtract 8 from both sides

and

Divide by 2 to both sides to get the answers

and

Plug each answer back into the original.

and

Both answers check out correctly.

This question is slightly different because the x variable is in the absolute value and the absolute value is not alone on one side of the equation. The first step is to get the absolute value alone by moving everything else on that side of the equation to the other side.

In this problem, that positive 5 is in the way so move it by subtracting 5 from both sides of the equation.

Now, because the absolute value is by itself on one side of the equation you can set it equal to the positive and negative value of the other side. You should have

and

Get x alone by first adding 9 to both sides of each equation

and

Divide by 4 from both sides of each equation for your answers, making sure to reduce the fraction you get in the second answer to its lowest terms.

and

Plug back in the original to check

and

Both answers successfully check out

The absolute value is alone on one side of the equation and ready to be set equal to the positive and negative of the other side of the equation.

and

It is easier to remove the divide by 3 from the left side of the equation first by multiplying both sides of the equations by 3

and

Then add 8 to both sides to get x by itself

and

Plug back into the original to check as you always should.

and

Two correct answers!

If you look at this problem too quickly, you may see an absolute value with the other side being a negative and immediately think no solution. However, the absolute value is not alone on the left side. It still has that negative sign in front, which means you need to either multiply or divide it out from both sides before solving.

Dividing out the negative will get you

Now, this looks similar to other problems you have seen. Set the absolute value equal to the positive and negative value of the other side

and

Subtract 12 from both sides of each equation

and

Put back into the original to check your work.

and

Once again, the absolute value is not alone on one side of the equation so the first step is making that happen. When a number is in front of an absolute value it is understood as multiplication similar to when a number is in front of a parentheses. Step one is to divide 5 from both sides to get the absolute value alone on the left of the equation.

Now, since the absolute value is alone on one side, set the other side equal to both its positive and negative and drop the absolute value signs.

 and

Subtract 10 from both sides of each equation

and

Put each of these back in the original to double check your answer

and

These answers are both correct.

Again, you need to be careful looking at the absolute value on this question. Notice the denominator is not in the absolute value. You have to move the 4 in the denominator to the other side of the equation. To do this multiplyboth sides by 4. You will then get.

Now since the absolute value is alone on one side of the equation you can remove the absolute value symbols and set the equation equal to both the positive and negative of the other side, which is 12 and -12.

and

Subtract 2 from both sides of the equations

and

Then divide by 4 on both sides of the equations to get x alone

 and 

Finally make sure to reduce to the fraction to lowest terms or change to a mixed number depending on your teacher's preferences

 or and  or

Plug back into the original (I will just be using the fractions not the mixed numbers but it is the same result)

and

Both answers check out

This question is no solution because once you get the absolute value alone on one side the other side is negative. The first step towards getting the absolute value alone is adding 11 to both sides

Then, divide by -2 to both sides

Now, you have an equation where the absolute value is alone on one side and the other side is negative. This is impossible so there is no solution.

If you had not seen this and accidentally solved for both the positive and negative values of the other side of the equation you would have

and

Subtract 4 from both sides to get x alone

and or

Plugging the fraction answers in the original equation you would see that neither of these answers work

and

No answers work. There is no solution.

This question is another one that may trick people into putting no solution if they are not looking carefully at the problem. First, get the absolute value alone on one side of the equation. Only then should you look at the other side of the equation to see if it is negative and without a solution. This is not the case in this problem.

Get the absolute value alone by adding 9 to each side of the equation

Now that the absolute value is alone on one side, drop the absolute value symbols and set it equal to the positive and negative 7 of the other side of the equation. You should have

and

Subtract 1 from both sides

and

Divide by 6 on both sides to get x alone

and   After reducing the fraction

Plug both back into the original

and

Both answers check out.

Sometimes when you do enough absolute value problems with variables inside of them, you forget how to do ones without a variable inside. In this case, you are only going to end up with one answer because you can immediately simplify and eliminate the absolute value.

No absolute value after simplifying means you are only going to have one answer, .

To solve this absolute value equation, we must first understand absolute values.

An absolute value is the distance a number is from zero on the number line. Because it is a distance, it cannot equal a negative number. 

However, the value inside the absolute value CAN be negative (once the absolute value is applied, the answer becomes positive).

So, given the equation,

we will have to create 2 separate equations without the absolute values (we just want to find the value of x located inside the absolute values).  One that is equal to a positive answer, and one that is equal to a negative answer.

and

So,

Then,

Therefore,  or .  Either of these answers would make the absolute value equation true.